arXiv subjects

Xiangdong Ye

Publications and source records attributed to Xiangdong Ye.

At least 19 recordsLinked to original sources

arXiv2026

Connectedness of polynomial diagonal orbit closures for minimal nilrotations and applications

For a minimal nilrotation on a compact connected nilmanifold, we prove that the polynomial diagonal orbit closure associated with any finite family of polynomials with integer coefficients vanishing at the origin is connected. This resolves a conjecture of Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson. Combined with their equivalence theorem, our result yields polynomial multiple recurrence in every prescribed residue class in topological dynamics, provided that the corresponding power of the transformation is minimal. Furthermore, we independently establish the measure-theoretic counterpart of this recurrence phenomenon. Finally, we construct a totally minimal nilsystem for which the lower central series identity proposed by Leibman fails.

math.DS↗

arXiv2026

MathCoPilot: An Interactive System for Human-AI Symbiotic Paradigm of Mathematical Research

Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini~3.1~Pro, GPT-5.4, and Claude~Opus~4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.

cs.AI↗

arXiv2026

Zero-Threshold Discrepancies for Multiple Correlation Sequences

We study the zero-threshold lifting problem for polynomial multiple correlation sequences with respect to the measure-theoretic pro-nilfactor. The structure theory for polynomial multiple averages implies that, at every positive threshold, positivity on the pro-nilfactor lifts to positivity in the original system, except on a set of zero upper Banach density. We demonstrate that this lifting property does not hold at the zero threshold. Specifically, we construct an ergodic system and two sets of positive measure for which the pro-nilfactor correlation is positive along a set of times with positive upper density, while the corresponding exact correlation vanishes on this set. This provides a negative answer to a question posed by Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson. %\cite{GKLMMRR}. Additionally, we prove a corresponding rigidity property. For any ergodic system, any essentially distinct family of integer polynomials vanishing at the origin, and any tuple of non-negative bounded functions, the zero-threshold discrepancy set is not piecewise syndetic.

math.DS↗

arXiv2026

Quasi-disjointness in topological dynamics

Motivated by Berg's notion of quasi-disjointness for ergodic systems, we introduce and investigate the concept of quasi-disjointness for minimal systems. Several equivalent characterizations are provided. We prove that quasi-disjointness is preserved under taking factors, proximal extensions, and group extensions. As a consequence, we establish that every minimal {\bf PI} system is quasi-disjoint from all minimal systems. In addition, some variant of quasi-disjointness, namely strong quasi-disjointness is also introduced and examined. Particularly, we prove that each {\bf AI} system is strongly quasi-disjoint from all minimal systems.

math.DS↗

arXiv2026

Mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields and applications

Let $λ$ be a self-dual Hecke character over a CM field $K$. Let $\mathfrak{p}$ be a degree one prime of the maximal totally real subfield $F$ of $K$ and $Γ_{\mathfrak{p}}$ the Galois group of the anticyclotomic $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. We prove that $$L(1,λν)\neq 0$$ for all but finitely many finite order characters $ν$ of $Γ_\mathfrak{p}$ such that $\varepsilon(λν)=+1$. For an ordinary prime $\ell$ with respect to the CM quadratic extension $K/F$, we also determine the $\ell$-adic valuation of the normalised Hecke $L$-values $L^{alg}(1,λν)$. As an application, we complete Hsieh's proof of Eisenstein congruence divisibility towards the CM Iwasawa main conjecture over $K$. Our approach and results complement the prior work initiated by Hida's ideas on the arithmetic of Hilbert modular Eisenstein series, studied via mod $\ell$ analogue of the André--Oort conjecture. The previous results established the non-vanishing only for infinitely many characters $ν$. Our approach is based on the arithmetic of a CM modular form on a Shimura set, studied via arithmetic of the CM field and Ratner's ergodicity of unipotent flows.

math.NT↗

arXiv2026

On subsets of integers having dense orbits

Let $A\subset \mathbb{N}$. We say $A$ is an $R$-sequence for a given minimal system $(Y,S)$ if there is $y\in Y$ such that $\{S^ny:n\in A\}$ is dense in $Y$. Richter asked if $A$ is an $R$-sequence for all minimal equicontinuous systems implies that $A$ is an $R$-sequence for all minimal systems. In this paper, we investigate this question and related issues within the framework of totally minimal systems, including a characterization of transitive systems that are disjoint from all totally minimal systems. A dynamical system is scattering (resp. weakly scattering) if its product with any minimal (resp. minimal and equicontinuous) system is transitive. It turns out that $(X,T)$ is scattering if and only if for any transitive point $x\in X$ and any minimal system $(Y,S)$ there is $y\in Y$ such that the orbit of $(x,y)$ is dense in $X\times Y$ if and only if for each transitive point $x\in X$ and any non-empty open subset $U$ of $X$, $\{n\in \mathbb{N}:T^nx\in U\}$ is an $R$-sequence. By combining this result with earlier work of Huang and Ye, we deduce that if scattering and weak scattering are distinct properties, then both Richter's question and Katznelson's question admit negative answers.

math.DS↗

arXiv2025

On systems disjoint from all minimal systems

Recently, Górska, Lemańczyk, and de la Rue characterized the class of automorphisms disjoint from all ergodic automorphisms. Inspired by their work, we provide several characterizations of systems that are disjoint from all minimal systems. For a topological dynamical system $(X,T)$, it is disjoint from all minimal systems if and only if there exist minimal subsets $(M_i)_{i\in\mathbb{N}}$ of $X$ whose union is dense in $X$ and each of them is disjoint from $X$ (we also provide a measure-theoretical analogy of the result). For a semi-simple system $(X,T)$, it is disjoint from all minimal systems if and only if there exists a dense $G_δ$ set $Ω$ in $X \times X$ such that for every pair $(x_1,x_2) \in Ω$, the subsystems $\overline{\mathcal{O}}(x_1,T)$ and $\overline{\mathcal{O}}(x_2,T)$ are disjoint. Furthermore, for a general system a characterization similar to the ergodic case is obtained.

math.DS↗

arXiv2024

Completely Li--Yorke chaotic homeomorphisms with positive entropy

It is an open problem whether a homeomorphism on a compact metric space satisfying that each proper pair is either positively or negatively Li--Yorke, called completely Li--Yorke chaotic, can have positive entropy. In the present paper, an affirmative answer to this question is given. In fact, for each homeomorphism $T$ with positive entropy such that each proper pair is not two-sided asymptotic, a completely Li--Yorke chaotic homeomorphism with positive entropy associated with the given homeomorphism can be constructed.

math.DS↗

arXiv2024

Veech's theorem of higher order

For an abelian group $G$, $\vec{g}=(g_1,\ldots,g_d)\in G^d$ and $ε=(ε(1),\ldots,ε(d))\in \{0,1\}^d$, let $\vec{g}\cdot ε=\prod_{i=1}^{d}g_i^{ε(i)}$. In this paper, it is shown that for a minimal system $(X,G)$ with $G$ being abelian, $(x,y)\in \mathbf{RP}^{[d]}$ if and only if there exists a sequence $\{\vec{g}_n\}_{n\in \mathbb{N}}\subseteq G^d$ and points $z_ε\in X,ε\in \{0,1\}^d$ with $z_{\vec{0}}=y$ such that for every $ε\in \{0,1\}^d\backslash\{ \vec{0}\}$, \[ \lim_{n\to\infty}(\vec{g}_n\cdotε)x= z_ε\quad \mathrm{and} \quad \lim_{n\to\infty}(\vec{g}_n\cdotε)^{-1}z_{\vec{1}}=z_{\vec{1}-ε}, \] where $\mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$.

math.DS↗

arXiv2024

Multiple recurrence without commutativity

We study multiple recurrence without commutativity in this paper. We show that for any two homeomorphisms $T,S: X\rightarrow X$ with $(X,T)$ and $(X,S)$ being minimal, there is a residual subset $X_0$ of $X$ such that for any $x\in X_0$ and any nonlinear integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, there is some subsequence $\{n_i\}$ of $\mathbb Z$ with $n_i\to \infty$ satisfying $$ S^{n_i}x\to x,\ T^{p_1(n_i)}x\to x, \ldots,\ T^{p_d(n_i)}x\to x,\ i\to\infty.$$

math.DS↗

arXiv2024

Disjointness with all minimal systems under group actions

Let $G$ be a countable discrete group. We give a necessary and sufficient condition for a transitive $G$-system to be disjoint with all minimal $G$-systems, which implies that if a transitive $G$-system is disjoint with all minimal $G$-systems, then it is $\infty$-transitive, i.e. $(X^k,G)$ is transitive for all $k\in\N$, and has dense minimal points. In addition, we show that any $\infty$-transitive $G$-system with dense distal points are disjoint with all minimal $G$-systems.

math.DS↗

arXiv2024

A counterexample on multiple convergence without commutativity

It is shown that there exist a probability space $(X,{\mathcal X},μ)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},μ)$ with $h_μ(X,T)=h_μ(X,S)=0$, and $f, g \in L^\infty(X,μ)$ such that the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{n}x)g(S^{n}x) \end{equation*} does not exist in $L^2(X,μ)$.

math.DS↗

arXiv2024

A refined saturation theorem for polynomials and applications

For a dynamical system $(X,T)$, $d\in\mathbb{N}$ and distinct non-constant integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, the notion of regionally proximal relation along $C=\{p_1,\ldots,p_d\}$ (denoted by $RP_C^{[d]}(X,T)$) is introduced. It turns out that for a minimal system, $RP_C^{[d]}(X,T)=Δ$ implies that $X$ is an almost one-to-one extension of $X_k$ for some $k\in\mathbb{N}$ only depending on a set of finite polynomials associated with $C$ and has zero entropy, where $X_k$ is the maximal $k$-step pro-nilfactor of $X$. Particularly, when $C$ is a collection of linear polynomials, it is proved that $RP_C^{[d]}(X,T)=Δ$ implies $(X,T)$ is a $d$-step pro-nilsystem, which answers negatively a conjecture in \cite{5p}. The results are obtained by proving a refined saturation theorem for polynomials.

math.DS↗

arXiv2023

A counterexample on polynomial multiple convergence without commutativity

It is shown that for polynomials $p_1, p_2 \in {\mathbb Z}[t]$ with ${\rm deg}\ p_1, {\rm deg}\ p_2\ge 5$ there exist a probability space $(X,{\mathcal X},μ)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},μ)$ with $h_μ(X,T)=h_μ(X,S)=0$, and $f, g \in L^\infty(X,μ)$ such that the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1} f(T^{p_1(n)}x)g(S^{p_2(n)}x) \end{equation*} does not exist in $L^2(X,μ)$, which in some sense answers a question by Frantzikinakis and Host.

math.DS↗

arXiv2023

Topological dynamical systems induced by polynomials and combinatorial consequences

Let $d\in {\mathbb N}$ and $p_i$ be an integral polynomial with $p_i(0)=0$, $1\le i\le d$. It is shown that if $S$ is piecewise syndetic in $\mathbb Z$, then $$\{(m,n)\in{\mathbb Z}^2: m+p_1(n),\ldots,m+p_d(n)\in S\}$$ is piecewise syndetic in ${\mathbb Z}^2$, which extends the result by Glasner and Furstenberg for linear polynomials. Our result is obtained by showing the density of minimal points of a dynamical system of ${\mathbb Z}^2$ action associated with the piecewise syndetic set $S$ and the polynomials $\{p_1,\ldots,p_d\}$. Moreover, it is proved that if $(X,T)$ is minimal, then for each non-empty open subset $U$ of $X$, there is $x\in U$ with $\{n\in {\mathbb Z}: T^{p_1(n)}x\in U, \ldots, T^{p_d(n)}x\in U\}$ piecewise syndetic.

math.DS↗

arXiv2023

Polynomial Furstenberg joinings and its applications

In this paper, a polynomial version of Furstenberg joining is introduced and its structure is investigated. Particularly, it is shown that if all polynomials are non-linear, then almost every ergodic component of the joining is a direct product of an infinity-step pro-nilsystem and a Bernoulli system. As applications, some new convergence theorems are obtained. Particularly, it is proved that if $T$ and $S$ are ergodic measure preserving transformations on a probability space $(X,{\mathcal X},μ)$ and $T$ has zero entropy, then for all $c_i\in {\mathbb Z}\setminus \{0\}$, all integral polynomials $p_j$ with $°{p_j}\ge 2$, and for all $f_i, g_j\in L^\infty(X,μ)$, $1\le i\le m$ and $1\le j\le d$, $$\lim_{N\to\infty} \frac{1}{N}\sum_{n=0}^{N-1}f_1(T^{c_1n}x)\cdots f_m(T^{c_mn}x)\cdot g_1(S^{p_1(n)}x)\cdots g_d(S^{p_d(n)}x),$$ exists in $L^2(X,μ)$, which extends the recent result by Host and Frantzikinakis. Moreover, it is shown that for an ergodic measure-preserving system $(X,{\mathcal X},μ,T)$, a non-linear integral polynomial $p$ and $f\in L^\infty(X,μ)$, the Furstenberg systems of $\big(f(T^{p(n)})x\big)_{n\in {\mathbb Z}}$ are ergodic and isomorphic to direct products of infinite-step pro-nilsystems and Bernoulli systems for almost every $x\in X$, which answers a problem by Frantzikinakis.

math.DS↗

arXiv2022

On structure theorems and non-saturated examples

For any minimal system $(X,T)$ and $d\geq 1$ there is an associated minimal system $(N_{d}(X), \mathcal{G}_{d}(T))$, where $\mathcal{G}_{d}(T)$ is the group generated by $T\times\cdots\times T$ and $T\times T^2\times\cdots\times T^{d}$ and $N_{d}(X)$ is the orbit closure of the diagonal under $\mathcal{G}_{d}(T)$. It is known that the maximal $d$-step pro-nilfactor of $N_d(X)$ is $N_d(X_d)$, where $X_d$ is the maximal $d$-step pro-nilfactor of $X$. In this paper, we further study the structure of $N_d(X)$. We show that the maximal distal factor of $N_d(X)$ is $N_d(X_{dis})$ with $X_{dis}$ being the maximal distal factor of $X$, and prove that as minimal systems $(N_{d}(X), \mathcal{G}_{d}(T))$ has the same structure theorem as $(X,T)$. In addition, a non-saturated metric example $(X,T)$ is constructed, which is not $T\times T^2$-saturated and is a Toeplitz minimal system.

math.DS↗

arXiv2021

An alternative for minimal group actions on totally regular curves

Let $G$ be a countable group and $X$ be a totally regular curve. Suppose that $ϕ:G\rightarrow {\rm Homeo}(X)$ is a minimal action. Then we show an alternative: either the action is topologically conjugate to isometries on the circle $\mathbb S^1$ (this implies that $ϕ(G)$ contains an abelian subgroup of index at most 2), or has a quasi-Schottky subgroup (this implies that $G$ contains the free nonabelian group $\mathbb Z*\mathbb Z$). In order to prove the alternative, we get a new characterization of totally regular curves by means of the notion of measure; and prove an escaping lemma holding for any minimal group action on infinite compact metric spaces, which improves a trick in Margulis' proof of the alternative in the case that $X=\mathbb S^1$.

math.DS↗